11. Tower of Hanoi - Computer Science - University of Vermont

11. Tower of Hanoi 

Robert Snapp 

snapp@cs.uvm.edu 

Department of Computer Science 

University of Vermont 

Robert R. Snapp © 2012 11. Tower of Hanoi CS 32, Fall 2012 1 / 20

The Tower of Hanoi: Origin 

In the great temple at Benares, beneath the dome which marks the center 

of the world, rests a brass plate in which are fixed three diamond needles, 

each a cubit high and as thick as the body of a bee. On one of these 

needles, at the creation, God placed sixty-four discs of pure gold, the 

largest disk resting on the brass plate, and the others getting smaller and 

smaller up to the top one. This is the Tower of Bramah. Day and night 

unceasingly the priests transfer the discs from one diamond needle to 

another according to the fixed and immutable laws of Bramah, which 

require that the priest on duty must not move more than one disc at a time 

and that he must place this disc on a needle so that there is no smaller disc 

below it. When the sixty-four discs shall have been transferred from the 

needle on which at the creation God placed them to one of the other 

needles, tower, temple, and Brahmins alike will crumble into dust, and with 

a thuderclap the world will vanish. 

(From, De Parville, La Nature, 1884, part I, pp. 285–286; as translated in W. W. Rouse Ball and 

H. S. M. Coxeter, Mathematical Recreations and Essays, 13th edition, Dover, NY, 1987, 

p. 317.) 


The Tower of Hanoi: The Puzzle 

Initial State: 

Goal State: 

Only one disk can be moved at a time, from one peg to another, such that it never is 

placed on a disk of smaller diameter. 


Tower of Hanoi: One Disk Solution 

1 Move. 


Two Disk Solution 

Note that we perform the solution to the one-disk Tower of Hanoi twice (once in the 

top row, and once in the bottom row). Between rows, we move the bottom disk. Thus, 

we require 

2 1 C 1 D 3 Moves 


Tower of Hanoi: Three Disk Solution 

Note that we perform the solution to the two-disk Tower of Hanoi twice (once in the 

top row, and once in the bottom row). Between rows we move the bottom disk. Thus, 

we require 

2.2 1 C 1/ C 1 D 7 Moves 


Tower of Hanoi: What we know so far 

Let M.n/ denote the minimum number of legal moves required to complete a tower 

of Hanoi puzzle that has n disks. 

n M.n/ 

1 1 

2 3 

3 7 

Following the pattern, for n D 4 we need to solve the three-disk puzzle twice, plus 

one more operation to move the largest disk. Thus, 

M.4/ D 2 M.3/ C 1 D 2 7 C 1 D 15: 

Similarly, for n D 5 disks, we expect that we will need to perform 

M.5/ D 2 M.4/ C 1 D 2 15 C 1 D 31: 


Tower of Hanoi: n Disk Analysis 

Let M.n/ denote the minimum number of legal moves required to complete a tower 

of Hanoi puzzle that has n disks. 

Before the largest disk (i.e., the n-th disk) can be moved to the rightmost peg, all 

of the remaining (n 1) disks must moved to the center peg. (These n 1 disks 

must be somewhere, and they can’t obstruct the transfer of the largest disk.) 

This requires M.n 1/ legal moves. 

It takes 1 more operation to move the n-th disk to the rightmost peg. 

Finally, another legal sequence of M.n 1/ steps is required to move the n 1 

disks from the center peg, to the rightmost peg. 

We thus obtain the recursion relation, 

M.n/ D 2M.n 1/ C 1: 


Tower of Hanoi: Solution 

With the solution for a single disk 

the recursion relation 

defines the solution 

M.1/ D 1 

M.n/ D 2M.n 1/ C 1: 

M.n/ D 2 n 

In are hierarchy of algorithms, this would be called exponential or O.2 n /. 

Robert R. Snapp © 2012 11. Tower of Hanoi CS 32, Fall 2012 9 / 20 

1:

Tower of Hanoi: Practical Consequences 

The practical difficulty with exponential algorithms is that they can quickly grow out of 

hand. (With each additional disk, the minimum number of operations essentially 

doubles.) N.B., 1 century 4:5 10 9 seconds. (The age of the universe is 4 10 17 

seconds.) 

n 2 n 1 n 2 n 1 n 2 n 1 

8 255 19 524287 30 1073741823 

9 511 20 1048575 31 2147483647 

10 1027 21 2097151 32 4294967295 

11 2047 22 4194303 33 8589934591 

12 4095 23 8388607 34 17179869183 

13 8191 24 16777215 35 34359738367 

14 16383 25 33554431 36 68719476735 

15 32767 26 67108863 37 137438953471 

16 65635 27 134217727 38 274877906943 

17 131071 28 268435455 39 549755813887 

18 262143 29 536870911 64 18446744073709551615 


Tower of Hanoi: Strategy 

We have seen that the solution to this puzzle is recursive: 

In order to move all n disks to the right peg, we must first move the top n 1 

disks to the center peg. 

Before this, we must move the top n 2 disks to the right peg. 

And before this, we must move the top n 3 disks to the center peg, and so on. 

So what should the first move be? Should the top disk be moved to the center peg, or 

to the right peg? 


Tower of Hanoi: Solution Path 

If the number of disks is odd, then the first move should be to transfer the top 

disk to the right peg. 

If the number of disks is even, then the first move should be to transfer the top 

disk to the center peg. 

The sequence of states that is visited in the course of solving the puzzle is called the 

solution path. 

The length of the shortest solution path to the n-disk puzzle is 2 n . 


Tower of Hanoi: Legal States 

A configuration of disks in the Tower of Hanoi puzzle is said to be a legal state if no 

disk rests on a disk of smaller diameter. That is, the largest disk on each peg must be 

placed on the bottom, and the remaining disks must be placed in the order of 

decreasing diameter. 

Question: How many legal states are there for n disks placed on three pegs? 


Tower of Hanoi: Accessible States 

A configuration of disks is said to be anaccessible state, if it can be realized from the 

initial state after a legal sequence of moves. 

Question: How many accessible states are there for n disks placed on three pegs? 


Tower of Hanoi: Examples 

Number Length of Number Fraction of 

of Disks Solution Path of Legal States Visited States 

n 2 n 3 n .2=3/ n 

1 2 3 0.6666667 

2 4 9 0.4444446 

3 8 27 0.2962963 

4 16 81 0.1975309 

5 32 243 0.1316872 

6 64 729 0.0877915 

7 128 2187 0.0585277 

8 256 6561 0.0390184 

32 4294967296 1853020188851841 2:32 10 6 


State Representation 

It’s cumbersome to represent a state by an illustration. Instead we will adopt a list 

notation. 

For example, for a five disk puzzle, the notation 

indicates 

(2 1 3 1 1) 

The smallest disk is on peg 2 (the center peg). 

The next larger disk is on peg 1 (the left peg). 

The next larger disk is on peg 3 (the right peg). 

The next larger disk is on peg 1. 

The next larger (i.e., largest) disk is on peg 1. 

The only legal configuration that describes this is 

1 2 3 


State Graph: 1 disk 

(2) 

(1) 

Each state is represented by a labeled vertex; legal moves are represented by edges. 

Robert R. Snapp © 2012 11. Tower of Hanoi CS 32, Fall 2012 17 / 20 

(3)

State Graph: 2 disks 

(23) 

(21) 

(11) 

(31) 

(32) 

(33) (13) (12) (22) 


State Graph: 3 disks 

(223) 

(221) 

(111) 

(311) (211) 

(321) 

(121) (131) 

(231) 

(331) 

(332) 

(123) (323) (232) (132) 

(313) (212) 

(133) (122) 

(333) (222) 

(233) (213) (113) (112) (312) (322)

11. Tower of Hanoi - Computer Science - University of Vermont

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